Howdy, tgtmkr!
Good of you to show some caution in shooting that new .22 Hornet barrel on your H&R Topper 20 gauge. Caution and common sense: that is is how YOUNG shooters get to become O|LD shooters! *GRIN*
The break-open single shot H&R Model 158 "Topper" was made in several combo/interchangeable barrel styles, including .45-70, .30-30, and your .22 Hornet, as well as shotguns, of course.
Take an empty, sized, and primed .22 Hornet case. No powder, no bullet. It's best to have a reloader with the proper tools prepare this case for you, but lacking that, you can (CAREFULLY) taeka loaded cartridge and pull the bullet using vise-grip pliers and and empty the powder.
Load the primed case, and fire it. Examine the primer dent and amount the primer protrudes from the base of the case. Does it look normal? If you have feeler gauges, measure it. You can do this on some very flat surface, like a sheet of plate glass, or a machine table top (table saw, milling machine, etc.)
If the primer protrusion is no more than about 0.006 or 0.008" inch (six to 8 thousandths of an inch), it is safe to fire.
H&R chambered their barrels so the base of the cartridge is flush with the end of the barrel. If your headspace checks within those measurements, it's SAFE!
HTH
John